Section 4 - Spatial dependence - Looking at causes and effects in a geographical context:

• Spatial autocorrelation - what is it, how to measure it with a GIS.

• The independence assumption and what it means for modeling spatial data.

• Applying models that incorporate spatial dependence - tools and applications.

Spatial dependence

• what happens at one place depends on events in nearby places

• all things are related but nearby things are more related than distant things (Tobler's first law of geography)

• positive spatial dependence:
• • nearby things are more alike than things are in general

• negative spatial dependence:
• • nearby things are less alike than things are in general
  • conceptual problems with negative spatial dependence
  • • e.g. the chessboard

• spatial autocorrelation measures spatial dependence
• • an index, rather than a parameter of a process
  • dependence between discrete objects, or dependence in a continuous field?

• a world without positive spatial dependence would be an impossible world
• • impossible to map
  • impossible to describe, live in
  • hell is a place with no spatial dependence

• compares the squared differences in value between neighboring objects with overall variance in values

• calculates the product of values in neighboring objects

• related to Geary but not in a simple algebraic sense

Calculation of the Geary index of spatial autocorrelation

• i.e. neighboring values are slightly more similar than one would expect if the values were randomly allocated to the four areas

• see the discussion of variograms and Kriging

• the term geostatistics is normally associated with continuous space, spatial statistics more with discrete space
•  

• Idrisi calculates autocorrelation over a raster

• code has been written to calculate autocorrelation in ARC/INFO (see NCGIA Technical Paper 91-5)

• contact Dan Griffith, Syracuse University; Luc Anselin, University of Illinois

• some of these fail to take advantage of GIS capabilities, for generating input data and displaying output

• see also Spacestat
•  

• suppose there is a relationship between number of AIDS cases and number of people living in an area

• the form of this relationship will vary spatially
• • in some areas the number of cases per capita will be higher than in others
  • we could map the constant of proportionality

• spatial heterogeneity describes this geographic variation in the constants or parameters of relationships

• when it is present, the outcome of an analysis depends on the area over which the analysis is made
• • often this area is arbitrarily determined by a map boundary or political jurisdiction

Geographically weighted regression (GWR)

fits a model such as y = a + bx

but assumes that the values of a and b will vary geographically

determines a and b at any point by weighting observations inversely by distance from that point

diagram

Geographical brushing:

• a technique of ESA

• a user-defined window is moved over the map

• analysis occurs only within the window
•  

•  
• e.g. regression analysis assumes that the observations (cases) are statistically independent

• this violates the first law of geography

• in general, analysis in space is very different from conventional statistical analysis (although this is very often carried out on spatial data)
•  

• the relationship between land devoted to growing corn and rainfall in a Midwestern state like Kansas

• rainfall available at 50 weather stations

• percent of land growing corn available for 100 counties

• use a method of spatial interpolation to estimate rainfall in each county from the weather station data

• plot one variable against the other, and perhaps fit a regression equation

• how many data points are there?
• • the more data points, the more significant the results
  • 100 (the number of counties)?
  • 50 (the real number of weather observations)?
  • something in between?

• more data points can be invented by intensifying the sample network using spatial interpolation, but no more real data has been created by doing so

• both variables are strongly spatially autocorrelated, violating an assumption of regression

• the significance of the analysis is now uncertain

• methods of spatial regression try to overcome this problem in a systematic way
• • see Spacestat

An example

Crime and income in Los Angeles

rate of car thefts (per sq km per year)

median annual income in thousands

per census tract

5,000 observations

b = increase in car thefts per sq km per thousand dollars median income
  = -0.22

R² = percentage of variation in car thefts explained by income
   = 0.26

is this significant?

is it significant at the 95% level of confidence?

in a population of millions of census tracts, exhibiting the same range of rates of car thefts and median incomes, but no relationship between them (b = 0, R² = 0), could a sample of 5,000 census tracts have exhibited the same
degree of apparent relationship, or more, purely by chance?

but, but, but...we don't have a random sample of a larger population

there are only 5,000 tracts in LA and we have all there is

A related issue - the MAUP
 

• many statistics are reported by averaging or summing over polygons - e.g. populations of counties, average elevation

• it is commonly necessary to interpolate such values to new polygons which do not coincide
• • e.g. from census tracts with known populations to school districts
  • source zones have known populations
  • populations of target zones are unknown

• the best method of solving this problem is to create a continuous surface from the source data, then to integrate this surface to the new target areas

• density is constant within source zones

• density is constant within target zones

• density is constant within some third set of control zones

• density varies smoothly (Tobler's Pycnophylactic interpolation)

• e.g. Openshaw and Taylor

• 99 counties of Iowa

• two variables - % over 65, and % Republican
• • correlation for the counties was .3466
  •  

• e.g. 48 regions - correlations between -.548 and +.886

• e.g. 12 regions - correlations between -.936 and +.996

• evaluate the range?

• are we asking the right question?
• • is scale part of the question rather than a mere matter of implementation?